Volume 28 Issue 3 - December 31, 2014 PDF
System identification for a general class of observable and reachable bilinear systems
Cheh-Han Lee1, Jer-Nan Juang1,2,*
1 Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
2 Aerospace Engineering Department, Texas A & M University, USA
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Bilinear systems exist as a gateway between the worlds of linear systems and of nonlinear systems. A bilinear plant model constituting a state and control-input coupling term in addition to a linear term forms the simplest model of nonlinear systems. In spite of their simple form, bilinear systems are widely applied to engineering, biology and economics.

A special input sequence is designed for continuous-time bilinear system identification to a more general class of bilinear system models. Such bilinear systems are reachable and observable but not necessary true for their linear term of the bilinear model. Using the special input sequence, one can identify a bilinear system by only one experiment like the SEMP method[1,2], so the whole algorithm is called the single experiment with almost repeated input sequence (SEARIS) method. The SEARIS method keeps the concept of employing the free-decay response but abandons the use of the observability of the linear term of the bilinear model, which is the key component but also the drawback of the SEMP type methods.

The input sequence designated for the SEARIS method is somewhat complicated, so the details are neglected here. A simple example as shown in Figure 1 is used to explain the input sequence. As the figure displayed, the input sequence consists U1, U2, and U3, which are constituted by more basic units U0, W0, etc. Using such an input sequence with system theory and system identification techniques, one can identify the observable and reachable bilinear systems. Numerical examples show good correspondence (see figure 2) between the identified bilinear models and the original nonlinear systems.

Figure 1 Typical input sequence.

Figure 2 Measured (o) and simulated output (-).


  1. Juang, J.-N. and Lee, C.-H., Continuous-time Bilinear System Identification using Single Experiment with Multiple Pulses, Nonlinear Dynamics, Vol. 69, No. 3, pp. 1009-1021, 2012, DOI: 10.1007/s11071-011-0323-9.
  2. Lee, C.-H. and Juang, J.-N., Nonlinear System Identification - A Continuous-Time Bilinear State Space Approach,  The Journal of the Astronautical Sciences, Vol. 59, Nos. 1 & 2, pp. 409-431, January-June 2012.
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